Amazingly, this trick works. Due to improved algorithms for learning the score coming from the generative modeling, the applicability of this approach is very broad. I had spent several years making false starts on implementing the Malliavin calculus, but in the end, we found a route around it ;)

Jérémie Klinger found a simple trick back to Girsanov: you take the perturbation in the diffusion at the level of the Fokker-Planck equation and rewrite it to be included in the drift. The resulting drift then has a term proportional to \nabla \log \rho(x,t), what machine learners call the score!

The “classical” strategy for diffusion sensitivities comes from financial mathematics and is called the Malliavin calculus, it’s very explicit for simple models like Black Scholes but for a general Langevin equation, it is no easy feat to compute the sensitivity.

In many biological and active systems, diffusivity is highly spatially dependent, and the theory for perturbations in such cases is rather limited, largely based on beautiful work by Leticia Cugliandolo and work by Falasco and Baeisi, among many others.

Far from equilibrium, it is not so easy: one needs to understand the dynamics, and this requires working with dynamical trajectories and their associated path measures. Classically, we do this using the Girsanov theorem, which constructs a “relative path measure” as we perturb the drift term.

Computing response functions or “sensitives” requires understanding how an external perturbation drives the change in some observable. For equilibrium systems, Onsager taught us that this can be understood with correlation functions.

also I must say often when I read new methods being pre-printed, while I appreciate the eagerness to make a splash, many folks seem unaware of the long history of this field & its assessments - to their detriment

If in CADD, pls read through D3R's last paper
pubmed.ncbi.nlm.nih.gov/31974851/

Can verify that the code works, too :)

If you're talking about order in the dynamics itself, then dynamical large deviation theory is very relevant and the canonical review is by Hugo Touchette arxiv.org/abs/0804.0327

There's no easy way to do this in general, but computing the stationary distribution for a nonequilibrium dynamics might be a possible in some low dimensional system or systems with special structure ( ones where you can represent the distribution with tensor networks). Simulations otherwise...